Pierre de Fermat

French lawyer, amateur mathematician

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synonym Fermat (Fermat) generally refers to Pierre de Fermat

Pierre de Fermat (August 17, 1601 to January 12, 1665), France Lawyers and amateurs mathematician 。 His achievement in mathematics is no less than that of a professional mathematician. He seems to be right number theory Most interested in modern Calculus The establishment of has contributed. It is known as the "King of Amateur Mathematicians".

Chinese name: Pierre de Fermat
Foreign name: Pierre de Fermat
Alias: King of Amateur Mathematicians
Nationality: France
one's native heath: France
date of birth: August 17, 1601

Date of death: January 12, 1665
Occupation: lawyer
Representative works: Introduction to Plane and Three dimensional Trajectories 、 Method of finding the maximum and minimum
Key achievements: Fermat's big theorem
analysis geometry Basic principles of
Gender: male

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Pierre de Fermat is seventeenth century French lawyer and amateur mathematician. The reason why it is called amateur is that Pierre de Fermat has a full-time job as a lawyer. According to the actual pronunciation in French and referring to the English pronunciation, his surname is also often translated as "Ferma". Fermat's Last Theorem is used to be called Fermat's big theorem The original name of the western mathematical world "last" means that other guesses have confirmed that this is the last one. In his works written at the beginning of the 20th century, the famous mathematical historian E.T. Bell called Pierre de Fermat "the king of amateur mathematicians". Bell was convinced that Fermat was more successful than most professional mathematicians of Pierre de Fermat's contemporaries. The 17th century was an active century for outstanding mathematicians, and Bell believed that Fermat was the most prolific star among mathematicians in the 17th century.

Personal achievements

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Contribution to analytic geometry

Fermat is independent of Lenai· Descartes The basic principle of analytic geometry is discovered.

Before 1629, Fermat wrote about the ancient Greek geometers in the third century BC Apollonius Lost《 Plane trajectory 》One book. another use Algebra Method pair Apollonius Some lost proofs of the track have been supplemented, especially for the geometry of ancient Greece Apollonius conic theory Summarized and sorted out, and made a general study of the curve. He wrote an eight page paper in Latin in 1630《 Introduction to Plane and Solid Trajectories 》。

In 1636, Fermat and Mason, a mathematician at that time· Lobeval I began to write a letter and talked about my math work. However, the publication of Introduction to Plane and Three dimensional Trajectories was 14 years after Fermat's death. Therefore, before 1679, few people knew about Fermat's work, which was truly groundbreaking.

Introduction to Plane and Solid Trajectories records Fermat's find 。 He pointed out: "The one determined by two unknowns equation , corresponding to a track, a straight line or curve can be drawn. " Fermat's discovery was seven years before Lerney Descartes discovered the basic principle of analytic geometry. In his book, Fermat also studied the general linear and circular equation , and about hyperbola , ellipse parabola A discussion was held.

Descartes looks for its equation from a trajectory, while Fermat studies the trajectory from the equation, which are two relative aspects of the basic principles of analytic geometry.

In a letter in 1643, Fermat also talked about his idea of analytic geometry. He talked about cylinder 、 Elliptic paraboloid 、 Hyperboloid of two sheets And ellipsoid, pointing out that an equation with three unknowns represents a curved surface And made further research on it.

Contribution to calculus

One of Fermat's manuscripts in 1660

16. In the 17th century, calculus was the most brilliant pearl after analytic geometry. it is evident to anyone, Newton and Leibniz Is the founder of calculus, and there are at least tens of people before it scientist It laid a foundation for the invention of calculus. But among many pioneers, Fermat is still worth mentioning.

The tangent problem of curves and the maximum and minimum problem of functions are one of the origins of calculus. This work is relatively old and can be traced back to the ancient Greek period. Archimedes In order to find the area of any figure enclosed by a curve Exhaustion method 。 Due to the cumbersome and clumsy nature of exhaustion law, it was gradually forgotten, and was not valued again until the 16th century. because Johannes Kepler When exploring the laws of planetary motion, we encountered the problem of how to determine the area and arc length of an ellipse. The concepts of infinity and infinitesimal were introduced and replaced the tedious exhaustion method. Although this method is not perfect, it is self Kavalieri Mathematicians since they came to Fermat have opened up a very broad space for thinking.

Fermat set up a quest tangent , maximum and minimum, and definite integral method Calculus Has made significant contributions.

Contribution to probability theory

As early as in ancient Greece, the problem of contingency and inevitability and their relationship has aroused the interest and debate of many philosophers, but it is only after the 15th century that they were mathematically described and dealt with. In the early 16th century, Italy appeared Caldano And other mathematicians study the play a game Opportunity, explore the division of gambling money at the point of the game. In the 17th century, the French Pascal And Fermat studied Italian pachauri 's works《 abstract 》, established communication links, and thus established Probability The foundation of.

Fermat considers that there are 2 × 2 × 2 × 2=16 possible outcomes of four gambles. Except for one outcome, which means that the opponent wins all four gambles, the first gambler wins all the others. Fermat did not use the word probability at this time, but he came to the conclusion that the winning probability of the first gambler is 15/16, that is, the ratio of the number of favorable cases to the number of all possible cases. This condition can generally be satisfied in combination problems, such as card games, throwing silver and touching the ball from the jar. In fact, this research has laid the foundation of the game for the mathematical model of probability - the abstraction of probability space, although this summary was not made until 1933 Kolmogorov Made.

Fermat and Blaise· Pascal The concept of mathematical expectation, the basic principle of probability theory, was established in mutual communication and works. This starts from the mathematical problem of points: how to determine the division of gambling in an interrupted game between players who are supposed to have the same skills, and how to know the score of two players when they are interrupted and the score needed to win in the game. Fermat made a discussion like this: a player A needs 4 points to win, and player B needs 3 points to win, which is Fermat's solution to this special situation. Because it is obvious that at most four times can decide the outcome.

The concept of general probability space is the thorough axiomatization of people's intuitive ideas about concepts. From a purely mathematical point of view, the finite probability space seems to be bland. But once the random variable And mathematical expectations, they become a magical world. Fermat's contribution lies in this.

Contribution of logarithm theory

At the beginning of the 17th century, the ancient Greek mathematicians of the third century AD spread in Europe Difantu Written《 arithmetic 》One book. In 1621, Fermat bought this book in Paris Indefinite equation Have conducted in-depth research. Fermat limited the study of indefinite equations to integer Within the range, thus starting number theory This branch of mathematics.

Fermat's achievements in the field of number theory are enormous, including:

Fermat's big theorem : n ≥ 3 is an integer, then the equation x ^ n+y ^ n=z ^ n does not satisfy the integer solution of xyz ≠ 0. This is an indefinite equation, which has been developed by British mathematicians Wiles Proved (1995), the process of proving is quite hard!

Fermat's small theorem : a ^ p-a ∨ 0 (mod p), where p is a prime number and a is a positive integer, and its proof is relatively simple. In fact, it is a special case of Euler's theorem. Euler's theorem is: a ^ φ (n) - 1 ∨ 0 (mod n), a, n are positive integers, and φ (n) is euler function , representing the number of positive integers less than n coprime with n (its expression Euler It has been concluded that“ euler formula ”Found in this entry).

In addition:

(1) All greater than 2 prime number It can be divided into 4n+1 and 4n+3.

(2) Prime numbers in the form of 4n+1 can, and can only be expressed as two in one way Square number The sum of.

(3) No prime number in the form 4n+3 can be expressed as the sum of two squares.

(4) A prime number of the form 4n+1 can and can only be used as the hypotenuse of a right triangle whose right angled side is an integer; The square of 4n+1 is and can only be the hypotenuse of two such right triangles; Similarly, the m-power of 4n+1 is and can only be the hypotenuse of m right triangles.

(5) The area of a right triangle with rational side length cannot be a square number.

(6) The prime number of 4n+1 shape and its square can only be expressed as the sum of two squares in one way; Its 3rd and 4th power can only be expressed in two ways as the sum of two squares; The 5th and 6th power can only be expressed as the sum of two squares in three ways, and so on until infinity.

(7) Found a second pair Affinity number : 17296 and 18416.

In the 16th century, it was believed that there was only one pair of compatible natural numbers: 220 and 284. Some boring people even added superstition or mystery to affinity numbers, and made up many myths. It also publicized that the affinity number Magic , magic, astrology and divination have important roles and so on.

More than 2500 years after the birth of the first affinity number, the wheel of history turned to the seventeenth century. In 1636, Fermat, the "king of amateur mathematicians" in France, found the second pair of affinity numbers 17296 and 18416, rekindled the torch for finding affinity numbers, and found light in the darkness. Two years later, French mathematician Lerney, the "father of analytic geometry"· Descartes (Ren é Descartes) also announced on March 31, 1638 that the third pair of affinity numbers 9437056 and 9363584 had been found. Fermat and Descartes In two years, the silence of more than 2000 years has been broken, and the wave of searching for affinity numbers in the mathematical world has been aroused.

Contribution to optics

Fermat's outstanding contribution in optics is that Principle of minimum action , also called Principle of action in the shortest time 。 This principle has a long history. As early as in ancient Greece, Euclid I put forward The law of straight line propagation of light and Reflection law 。 Later, Helen revealed the theoretical essence of these two laws - light takes the shortest path. After several years, this law has gradually been expanded into a natural law, and has become a philosophical concept The more general conclusion that "nature acts in the shortest possible way" finally came out and affected Fermat. Fermat's cleverness lies in turning this philosophical idea into a scientific theory.

Fermat also discussed the case that the path of light in a point by point medium takes a minimal curve. Some problems are explained by the principle of minimum action. This has greatly encouraged many mathematicians. in especial Leonhard Oura Competing use Variational method Trick is to apply this principle to the extremum 。 This directly led to Lagrange The achievement of, gave the concrete form of the minimum action principle: particle The integral of the product of its mass, speed and the distance between two fixed points is a maximum and a minimum; That is, for the actual path taken by the particle, it must be maximum or minimum.

Growth experience

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Childhood life

Fermat (also translated as "Fermat") was born in southern France on August 17, 1601 Toulouse Nearby Beaumont de Lomani 。 His father, Dominic Fermat, opened a large leather shop in the local area, with quite rich industries, which made Fermat live in a rich and comfortable environment when he was young.

Fermat's father was respected by people because of his wealth and business ethics, and therefore won the title of local affairs adviser. But when Fermat was young, he did not feel much superior because of his rich family. Fermat's mother's name is Clare de Rogge , born in a noble robe. dominica Fermat's great wealth and Rogge's great nobility formed his extremely rich status.

Learning period

Fermat was educated by his uncle Pierre when he was a child, and he got a good enlightenment education. He developed a wide range of interests and hobbies, which also had an important impact on his character. It was not until he was 14 that Fermat entered Beaumont de Lomani Public School , after graduation Universite d'Orleans and University of Toulouse Study law.

In France in the 17th century, the most important profession for men was to be a lawyer. Therefore, it is fashionable for men to study law, which is also admired by people. Interestingly, France has created good conditions for those "prospective lawyers" who have assets but lack qualifications to become lawyers as soon as possible. In 1523, Francois I A special agency was set up to sell officials and titles to the public. This kind of official sale social phenomenon Once produced, it will be out of control in response to the needs of the times.

Officialdom career

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Selling official positions, on the one hand, catered to those who were rich, so that they could obtain official positions and improve their social status, on the other hand, it also improved the financial situation of the government. So by the 17th century, any official position except court officials and military officers could be bought and sold. Until today, the court clerk, notary, messenger and other positions have not completely got rid of the nature of business. Many middle-class people benefit from the special products of buying official positions in France, and Fermat is no exception. Before he graduated from college, Fermat bought the positions of "lawyer" and "senator" in Beaumont de Romagne. When Fermat returned to his hometown after graduation in 1631, he easily became a pawn Toulouse Parliament Members of Parliament.

Although Fermat did not lose his official position from entering the society until his death, and he has been promoted year by year, it is recorded that Fermat has no political achievements, and his ability to deal with officialdom is also very ordinary, let alone leadership. However, Fermat did not stop his promotion. After seven years as a member of the local council, Fermat was promoted to the investigating senator, an official position with the right to investigate and challenge the administration.

In 1642, there was an authoritative person, the Supreme Court adviser Brisias Recommend Fermat to enter Supreme Criminal Court and French Grand Council The main court, which gave Fermat a better chance of promotion in the future. In 1646, Fermat was promoted to the post of chief speaker of the parliament, and later served as a speaker Catholicism Chairman of the Alliance. Fei Ma's officialdom career has no outstanding achievements to be commended, but he never used his power to blackmail people, never accepted bribes, and was honest and open Incorruptible , won people's trust and praise.

Naturalized nobility

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Fermat's marriage made Fermat one of the Robed nobility Fermat married his uncle and cousin Louise de Rogge 。 Fermat, who was proud of his mother's aristocratic lineage, now simply added the symbol "de" to his name.

Fermat had three daughters and two men. Apart from his eldest daughter, Crawley, who was married, all four children made Fermat feel respectable. Two daughters became priests, and the second son became priests Femares The archbishop of. Especially the eldest son Clement Somer He not only inherited Fermat's public office and became a lawyer in 1665, but also sorted out Fermat's mathematical works. If Fermat's eldest son had not actively published Fermat's mathematical works, it is hard to say that Fermat could have such a significant impact on mathematics, because most of the papers were published by Fermat's eldest son after his death. In this sense, Samore It can also be called the successor of Fermat's career.

family background

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Fermat's father, Dominique Fermat, was a fur merchant and the second consul of Pomont Lomen. Claire de Long, Pierre de Fermat's mother, came from a family of judges. Ferma was born in August 1601 (baptized in Pomont Lomen on August 20), and her parents were determined to cultivate Pierre de Ferma as a local governor.

Pierre de Fermat was born in Duluth When he was 30 years old, he served as a petition committee member in the same place. He married Louise Long in the same year and had three daughters and two sons. One of his sons, Clement Samuel Fermat, became Pierre de Fermat's main assistant in scientific research. After Fermat's death, he collated and published Pierre de Fermat's work achievements. In fact, this publication is also the source of Fermat's Last. Theorem, which is known for a long time. Because his family was rich, his father specially hired two tutors for Pierre de Fermat to receive systematic education at home instead of going to school. Although Fermat was not a child prodigy, he was also quite clever. Fermat's father was more open-minded and did not favor his children, so Fermat studied very hard and did well in arts and science. However, Pierre de Fermat's favorite subject was mathematics.^[1]

In 1617, Fermat was preparing for college entrance examination. His father wanted him to study law. Fermat also liked this subject, so he accepted his father's arrangement without much controversy.

Death

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Fermat was in good health all his life, but nearly died in the plague in 1652. After the New Year's Day in 1665, Fermat began to feel that his body had changed, so he was suspended on January 10. The third day, Fermat died. Fermat was buried in Casteres Cemetery , later buried in Toulouse In the family cemetery.

Personal evaluation

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Fermat had never received special mathematical education in his life, and mathematical research was just a hobby. However, in the 17th century France No mathematician can match him: he is one of the inventors of analytic geometry; about Calculus The contribution of birth is second only to Isaac Newton Gottfried William Van Leibniz probability theory He is the main founder of, and the sole supporter of the number theory world in the 17th century. In addition, Fermat also made important contributions to physics. Fermat, a generation of mathematical genius, can be called the greatest mathematician in France in the 17th century.

Google Commemoration

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On August 17, 2011, Google's logo was updated again. This time, it was more interesting. The logo tag said: "I found a wonderful proof of this theorem, but it's too small to write down here." This time, Google's logo commemorates the 410th anniversary of the birth of Pierre de Fermat, the king of amateur mathematicians. In fact, the tag in the LOGO quotes his words: "I am sure I have found an excellent proof, but the blank space of the book is too narrow to write down."

Google's homepage is replaced with a graffiti logo commemorating Pierre de Fermat

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